Repeating the process includes the instruction to repeat the process. That is, the opponent will keep sacrificing permanents until he or she sacrifices a land.

5/1/2012

Repeating the process does not include targeting an opponent. Only one opponent will have to sacrifice permanents.

5/1/2012

If any abilities trigger while Tyrant of Discord's enters-the-battlefield ability is resolving, those abilities will wait until the enters-the-battlefield ability finishes resolving before going on the stack. Starting with the active player, each player puts his or her abilities on the stack in any order.

5/1/2012

If the target opponent can't sacrifice permanents (perhaps because of Sigarda, Host of Herons), the opponent will choose a permanent at random but not sacrifice it. The ability will then finish resolving.